1. R. Marbach and H. M. Heise, Calibration Modeling by Partial Least-Squares and Principal Component Regression and its Optimization Using an Improved Leverage Correction for Prediction Testing, Chemometrics and Intelligent Laboratory Systems 9, 45-63 (1990)
2. R. Marbach, On Wiener Filtering and the Physics Behind Statistical Modeling, Journal of Biomedical Optics 7, 130-147 (January 2002)
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The invention relates to methods for calibrating multichannel measurement instruments.
Multichannel instruments are instruments that measure a set of multiple input signals first, and then use an algorithm to generate a desired single output number from the measured values. The input signals can be from one multidimensional measurement, e.g., light absorbance values at different optical wavelengths, or from several one-dimensional measurements, e.g., the input set of [temperature, pressure, humidity]; or from any combination of these two. Multichannel instruments are used in many different industries and for many different applications, under varying names. They may also be called, e.g., xe2x80x9cmultivariatexe2x80x9d or xe2x80x9cmultidimensionalxe2x80x9d or xe2x80x9cmultiple parameterxe2x80x9d or xe2x80x9cmulti-pixelxe2x80x9d or xe2x80x9cbroad spectrumxe2x80x9d or xe2x80x9clarge-areaxe2x80x9d etc. The common characteristic is that the instruments must be calibrated before use, i.e., an algorithm must be programmed into the instruments that translates the multiple measured numbers into the desired single output number. Calibration is the process of determining that algorithm.
Today""s procedures for calibrating multichannel instruments are ineffective and inefficient. The prior art knows two different approaches, viz., the so-called physical and statistical calibration methods, which so far have been thought of as separate and not interrelated (a summary is given, e.g., in Reference [1]). The mathematical tools used in the two approaches are similar, because both are based on linear regression techniques, but the methods differ substantially in the type and amount of data that the user has to measure in order to perform the calibration. The physical method is relatively cheap and intuitively easy to understand, however, this method can only be applied in very simple measurement situations and the quality of the result is almost always inferior to the statistical result. The statistical method is generally preferred because, in principle, it works in all cases and it converges against a desired optimal result, however, it requires large amounts of calibration data and the cost associated with collecting that data is usually very high.
At first glance it appears that, since calibration methods have been in widespread use for several decades now, the process of calibration would be fully understood and hard to improve on any further. However, much to the surprise of this author himself, this is far from true and the sad reality is that large amounts of money are currently being wasted on ineffective and inefficient procedures to find good multichannel algorithms.
The majority of the procedures in use today are based on the statistical approach, which works as follows. During a dedicated calibration time period the instrument is used as intended in the later measurement application and a large number of data points are measured and saved into memory. If the instrument measures, say, k channels, then each data point consists of (k+1)-many numbers, namely the k channel readings plus the xe2x80x9ctruexe2x80x9d value of the desired output. The xe2x80x9ctruexe2x80x9d value is measured using an independent reference instrument that serves as a standard to the calibration. Eventually, after a sufficient number of data points have been collected, a hyper-plane is fitted through the data points using standard linear regression techniques and the parameters of the fit are used to program the algorithm. With the advent of the personal computer, activity on the subject has increased to vast proportions and has even spawned several new branches of science, e.g., xe2x80x9cbiometrics,xe2x80x9d xe2x80x9ceconometrics,xe2x80x9d or xe2x80x9cchemometrics;xe2x80x9d along with about a dozen new scientific journals, a dozen new research institutes, scores of university chairs, and thousands of professionals graduated in the field.
The statistical approach works but has significant disadvantages, including:
1. Calibration time periods are often excessively long in order to model all the noise that the instrument is likely to see in future use; this is especially true in high-precision applications where low-frequency noises with time constants on the order of days, weeks or even months must be modeled;
2. the calibration data set is often affected by a dangerous effect called xe2x80x9cspurious correlationxe2x80x9d (discussed below) which can render results useless and can be difficult to detect;
3. there is no way to effectively use a-priori knowledge about the physics of the measurement process to ease the task of calibration; instead, the calibration is purely xe2x80x9cstatisticalxe2x80x9d and always starts from scratch;
4. there is no way to effectively and quantitatively assess the effect of hardware or measurement process changes on the calibration; consequently, there is no quantitative feedback mechanism that would tell in advance, e.g., what the effect of a planned hardware change on the system performance would be; and there is also no way to easily xe2x80x9cmaintainxe2x80x9d or update an existing calibration to slight changes in the instrument hardware or measurement process;
5. there is no way to easily xe2x80x9cre-usexe2x80x9d an existing calibration for a new but similar application;
6. there are severe marketing problems because the results of statistical calibration are hard to interpret which, in turn, makes end users reluctant to buy, and depend on, a machine the inner workings of which they do not fully understand.
The reason behind all of these problems is that there is currently no understanding about the relationship between the statistical calibration process and the underlying physics of the measurement problem. As a result, even in the best case when enough effort has been spent and the statistical method has actually converged against the desired optimum result, users are still left in a situation where there is always, a feeling of distrust against the solution because it is not physically understood (xe2x80x9cWill it really continue to work in the future?xe2x80x9d) and a feeling that one should have collected even more data (xe2x80x9cCould I get better than this?xe2x80x9d).
The reason for the widespread use of the statistical approach, in spite of all the problems listed above, is simply that for many measurement problems there is no alternative. Also, there is the generally accepted fact that, if enough calibration data can be collected, then the statistical method somehow converges against an optimal result that can not be outperformed by other solutions. In some simple measurement situations, users shy away from the statistical approach and instead apply the so-called physical approach to calibration. In this method, the user tries to identify the multidimensional fingerprints of each and every physical effect in the multidimensional data space, one effect at a time, and then reconstruct each measured set of input signals as a weighted sum of the individual effects. Unfortunately, the physical approach only works in very simple situations, and even then the results are inferior to those one could have gotten from the statistical approach. Worse, if the measurement problem is a complicated mixture of many physical effects (which is the principal reason why most users decide to do a multichannel measurement in the first place) the physical approach breaks down and does not work at all (see, e.g. Reference [1]).
The new methods work, in short, by translating the difficult inverse-problem posed by the statistical method into simpler, forward-problem, xe2x80x9cphysicalxe2x80x9d measurements of the signal and the noise. This, in turn, allows significant reductions in the amount of data needed for calibration, while simultaneously providing full insight and interpretability of the result and consistently outperforming the quality of the statistical method. The methods disclose how to compute the optimal regression vector; how to update the optimal result to account for small changes in the noise; how to choose a xe2x80x9cgoodxe2x80x9d subset of channels for measurement; and how to quantify the noise contributions from the multichannel measurement and from the reference measurement individually. The methods are adapted to different situations, i.e., to different amounts of knowledge that the user may have about the signal and the noise. The more knowledge the user has, the more significant the savings can be.
The object of this invention is to overcome the disadvantages of the prior art listed above. The described methods significantly reduce the cost of multichannel calibration and simultaneously deliver optimal quality, consistently outperforming the statistical method on both price and quality. The new methods combine the best of both worlds, viz., the quality of the statistical method with the low cost and interpretability of the physical method. The new methods can be applied for both calibration and calibration maintenance and are adapted to different situations, i.e., to different states of knowledge that the user may have about the signal and the noise, so that maximum cost savings can be realized in different situations.
To give just one example of the amount of money that can be saved, take the chemometrics field. This field is concerned with the calibration of instruments that measure optical spectra at multiple wavelength channels to quantify, e.g., a chemical concentration in an industrial process control application. A conservative estimate by this author is that  greater than U.S.$ 150 million worth of RandD and non-recurring engineering expenses worldwide can be saved annually just in the chemometric process-control market segment. This figure only counts expenses-saved, i.e., potential revenue increases due to increased customer acceptance of calibration-based products are not included.